#438190

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6)(7,18)(8,22)(9,25)(10,16)(11,19)(12,17)(13,14)(15,29)(20,34)(21,24)(23,26)(27,33)(28,32)(30,35)(31,36);(1,2)(3,4)(5,7,12,22,10,14,11,21)(6,13,17,24,16,18,19,8)(9,26,29,35,27,20,28,36)(15,31,33,23,32,30,25,34);(5,8,11,13,10,24,12,18)(6,14,17,22,16,7,19,21)(9,23,28,30,27,34,29,31)(15,26,33,36,32,20,25,35);(1,3,4,2)(5,9,10,27)(6,15,16,32)(7,20,14,26)(8,23,24,34)(11,28,12,29)(13,30,18,31)(17,33,19,25)(21,35,22,36)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345386);